Mathematical Sciences: RUI: Spectral Problems for Toeplitz and Other Structured Matrices

当n较大时，如果指定H的任意n×n元所需的独立参数的数目近似与n成比例，则H的方阵族H是有效构造的。Toeplitz、Hankel和Toeplitz-Plus-Hankel矩阵是有效构造矩阵的例子，正如T.Kailath和他的合著者定义的那样，具有给定位移秩的矩阵也是如此。这项研究的目的是：(A)发展求一般有效结构厄米特矩阵的单个特征值-特征向量对的快速算法，类似于主要研究者对Toeplitz和Toeplitz-plus-Hankel矩阵的算法；(B)研究Toeplitz矩阵的谱性质(特别是实对称Toeplitz矩阵的奇偶谱之间的关系)；(C)研究设计求非对称Toeplitz矩阵的实特征值的快速算法的可能性；以及(D)考虑Toeplitz-Sturm矩阵的谱性质。高效结构的厄米矩阵出现在许多重要的物理和科学应用中，包括涉及基于对随机变量的过去值的观测来预测随机变量的未来值的统计问题、信号处理(从传输的信号中提取信息的科学)以及地球物理应用，包括研究地球的自由振荡和地震等地震现象。在研究这些问题时，通常需要确定非常高阶(以数千计)的有效结构矩阵的单个特征值。拟议研究的主要目标之一是开发数值方法，通过其复杂性(计算要求)与矩阵阶数的二次幂成正比，而不是与阶数的三次方成正比的方法来寻找这些特征值，而不是像标准计算方法那样，不利用矩阵的特殊结构。

A family H of square matrices is efficiently structured if the number of independent parameters required to specify an arbitrary n by n member of H is approximately proportional to n when n is large. Toeplitz, Hankel, and Toeplitz-plus-Hankel matrices are examples of efficiently structured matrices, as are matrices of a given displacement rank, as defined by T. Kailath and his coauthors. The purposes of the proposed research are (a) to develop fast algorithms for finding individual eigenvalue-eigenvector pairs for general classes of efficiently structured Hermitian matrices, analogous to the principal investigator's algorithms for Toeplitz and Toeplitz-plus-Hankel matrices; (b) to study spectral properties of Toeplitz matrices (especially, the relationship between the even and odd spectra of real symmetric Toeplitz matrices); (c) to investigate the possibility of devising fast algorithms for finding the real eigenvalues of nonsymmetric Toeplitz matrices; and (d) to consider the spectral properties of Toeplitz-Sturm matrices. Efficiently structured Hermitian matrices arise in many important physical and scientific applications, including statistical problems involving prediction of future values of a random variable based on observations of its past values, signal processing (the science of extracting information from transmitted signals), and geophysical applications including the study of free oscillations of the Earth and seismic phenomena such as earthquakes. In studying these problems it is often necessary to determine individual eigenvalues of efficiently structured matrices of very high order (in the thousands). One of the main objectives of the proposed research is to develop numerical methods for finding these eigenvalues by methods whose complexity (computational requirements) is proportional to the second power of the order of the matrix, rather than to the third power of the order, as it is for standard computational methods that do not take advantage of the special structure of the matrix.